Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2006 | June | Q#9

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Question

Given that f(x)=( x2 – 6x)(x – 2) + 3x,

a.   express f(x) in the form x(ax2 + bx + c), where a, b and c are constants.

b.   Hence factorise f(x) completely.

c.   Sketch the graph of y = f(x), showing the coordinates of each point at which the graph meets the  axes.

Solution

a.
 

We are given;

b.
 

c.
 

We are required to sketch

We have seen in (a) that it can written as;

It is a cubic function.

ü Find the sign of the coefficient of . This gives the shape of the graph at the extremities.

ü Find the point where the graph crosses y-axis by finding the value of  when .

ü Find the point(s) where the graph crosses the x-axis by finding the value of  when . If  there is repeated root the graph will touch the x-axis.

ü Calculate the values of  for some value of . The is particularly useful in determining the  quadrant in which the graph might turn close to the y-axis.

ü Complete the sketch of the graph by joining the sections.

Sketch should show the main features of the graph and also, where possible, values where the  graph intersects coordinate axes.

First we find x-intercepts.

The point  at which curve (or line) intercepts x-axis, the value of . So we can find the  value of  coordinate by substituting  in the equation of the curve (or line).

We have found in (b) that given cubic function can be written as;

Therefore;

Now we have three options.

Therefore, the graph intersects x-axis at three points (0,0), (3,0) and (5,0).

Next we find y-intercept.

The point  at which curve (or line) intercepts y-axis, the value of . So we can find the  value of  coordinate by substituting  in the equation of the curve (or line).

Therefore;

Hence, the graph intercepts y-axis at (0,0).

desmos-graph (1).png

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